\documentclass[reqno]{amsart} \usepackage{graphicx} \usepackage{hyperref} \AtBeginDocument{{\noindent\small {\em Electronic Journal of Differential Equations}, Vol. 2007(2007), No. 120, pp. 1--11.\newline ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu \newline ftp ejde.math.txstate.edu (login: ftp)} \thanks{\copyright 2007 Texas State University - San Marcos.} \vspace{9mm}} \begin{document} \title[\hfilneg EJDE-2007/120\hfil Existence and uniqueness] {Solutions of fourth-order partial differential equations in a noise removal model} \author[Q. Liu, Z. Yao, Y. Ke \hfil EJDE-2007/120\hfilneg] {Qiang Liu, Zhengan Yao, Yuanyuan Ke} \address{Qiang Liu \newline Department of Mathematics, Jilin University, Changchun 130012, China} \email{matliu@126.com} \address{Zhengan Yao \newline Department of Mathematics, Sun Yat-Sen University, Guangzhou 510275, China} \email{mcsyao@mail.sysu.edu.cn} \address{Yuanyuan Ke \newline Department of Mathematics, Jilin University, Changchun 130012, China} \email{keyy@jlu.edu.cn (corresponding author)} \thanks{Submitted April 10, 2007. Published September 14, 2007.} \thanks{Supported by grants NNSFC-10531040, NNSFC-10471156, NSFGD-4009793 and \hfill\break\indent NSFGD-06300481} \subjclass[2000]{35K65, 35M10} \keywords{Existence; uniqueness; fourth-order; noise removal} \begin{abstract} In this paper, we discuss the existence and uniqueness of weak solutions for a fourth-order partial differential equation stemmed from image processing for noise removal. We also present some numerical tests for high order filters. \end{abstract} \maketitle \numberwithin{equation}{section} \newtheorem{theorem}{Theorem}[section] \newtheorem{lemma}[theorem]{Lemma} \newtheorem{corollary}[theorem]{Corollary} \newtheorem{definition}[theorem]{Definition} \section{Introduction} We study the fourth-order initial-boundary value problem \begin{gather} \frac{\partial u}{\partial t} + \frac{\partial ^2}{\partial x^2} \Phi'\big(\frac{\partial ^2 u}{\partial x^2}\big)= 0 \quad(x,t) \in Q_T, \label{tv1}\\ u(0,t) = u(1,t) = u'(0,t) = u'(1,t) = 0 \quad t\in(0,T),\label{bd1}\\ u(x,0) = u_0(x)\quad x\in I \label{bd3}, \end{gather} where $I = (0,1)$, $Q_T = I \times (0,T)$ and $\Phi: \mathbb{R} \to \mathbb{R^+}$ is an $N$ function; i.e. $\Phi(\cdot)$ is even, continuous, convex with $\Phi>0$ for $t>0$, \begin{equation}\label{cd1} \lim_{t\to 0}\frac{\Phi(t)}{t} \to 0 \quad\text{and}\quad \lim_{t\to \pm\infty}\frac{\Phi(t)}{|\,t|} \to + \infty. \end{equation} Here we assume that $\Phi$ satisfies the $\Delta_2 $-condition: \begin{equation}\label{cd2} \Phi(2\xi) \le K\Phi(\xi),\quad |\xi| \ge R, \end{equation} where $K>2$ and $R$ are two positive constants. In recent years, many nonlinear PDEs are proposed to deal with the trade-off between noise removal and edge preservation. Among them, the fourth-order parabolic PDEs have drawn great interest \cite{AP,CMM,ke,tai,you,XZ,XZ2}. Since they seek to minimize a cost functional which is an increasing function of the absolute value of the Laplacian of the image intensity function, they could decrease the staircasing property which may be undesirable under some circumstances \cite{AP,DSTC}. In general, the forms of fourth-order PDEs are analogous with the second order ones. For example, in \cite{you}, You and Kaveh proposed equation $$ u_t = -\Delta(g(\Delta u)\Delta u), $$ where $g(s) = {1}/{(1+s^2)}$, which is analogous with the Perona-Malik model \cite{pm}. In \cite{tai}, Lysaker \emph{et al} used the equation $$ u_t = -\Delta\big(\frac{\Delta u}{|\Delta u|}\big), $$ which is similar to TV model \cite{ROF92}. In \cite{SD}, Didas used the equation \eqref{tv1} with $\Phi(x) = 2\lambda \left(\sqrt{\lambda^2 - x^2}-\lambda \right)$, where $\lambda > 0$ and it is the Charbonnier filter \cite{GA}. Our model includes a class of more general equations \cite{GA,SD}, e.g. $\Phi(s) = \frac 1p |s|^p$, $p>1$. When $p=2$, a linear filter could be obtained. While this filter has very strong isotropic smoothing properties and does not preserve edges very well. One should then decrease $p$ in order to preserve the edges as much as possible, that is to say fast diffusion is desired. There are some other functions which satisfy the conditions \eqref{cd1} and \eqref{cd2}, for example: $$ \Phi(s) = |s|\ln(1+|s|), $$ and $$ \Phi(s) = |s|L_k(|s|), $$ where $L_i(s) = \ln(1+L_{i-1}(s))$ $(i= 1,2,\dots,k)$ and $L_0(s)= \ln (1+|s|)$, see \cite{FM,WZ}. Although the effectiveness of fourth order diffusion equations for noise removal has been proposed in \cite{AP,CH,CMM,ke,tai,you}, very little has been known about theoretical analysis. We refer to \cite{wzyl}, Chapter 4 for a nonlinear equation with double-degeneracy, \cite{travel} for traveling wave solutions in one dimension, \cite{BG} for the existence and uniqueness of \eqref{tv1} for $\Phi^\prime (s) = \arctan(s)$, \cite{ke} for the existence of a fourth order PDE by variational methods and \cite{XZ} for a generalized thin film equation. It is worth while mentioning that the initial data is chosen by the original image generally. We take the zero boundary value conditions for convenience, which corresponds to padding the boundary of the image with black. The plan of the paper is the following. In Section 2, we state some preliminaries and the main theorem. Section 3 is devoted to the proofs of our main results and Section 4 deals with some numerical experiments using finite difference methods by an explicit scheme. \section{Preliminaries and Main Result} In the following sections we always assume $\Phi(\cdot)$ is a function satisfied the condition \eqref{cd1} and \eqref{cd2}. Then the N-function $\Psi(\cdot)$ which conjugates to $\Phi(\cdot)$ is defined by $$ \Psi(s) = \sup_{t \in \mathbb{R}}\{t\cdot s - \Phi(t)\}. $$ We have the following Young's inequality, $$ s\cdot t \le \Phi(s) + \Psi(t). $$ For all $|s|>R$, we get (see \cite{Ad,WZ}) \begin{equation}\label{ine1} \Phi(s) \le \Phi'(s) s \le (K-1)\Phi(s) \end{equation} and \begin{equation}\label{ine2} 0 \le \Psi(\Phi'(s)) = \Phi'(s) s - \Phi(s) \le (K-2)\Phi(s). \end{equation} For any $s,t \in \mathbb{R}$, we have (see \cite{Ad,WZ}) \begin{equation}\label{ine3} \left(\Phi'(s) -\Phi'(t)\right)\cdot(s-t) \ge 0. \end{equation} \begin{lemma}[\cite{Ad,WZ}] \label{con0} If $\Psi$ conjugates to $\Phi$, then there exist positive numbers $p>1$, $R>0$, $R'>0$, $K_1>0$ and $K_2>0$ such that for all $s,t \in \mathbb{R}$, \begin{gather} \Phi(s) \le K_1 |s|^p, \quad |s| \ge R, \label{ine4} \\ \Psi(t) \ge K_2 |t|^{p'}, \quad |t| \ge R', \quad p' = \frac p{p-1}.\label{ine5} \end{gather} \end{lemma} \begin{lemma}[\cite{AF,WZ}] \label{con} Suppose $\{f_j\}\subset L^1(I;\mathbb{R})$ satisfies that $$ \int_I \Phi(f_j)dx \le C, $$ where $C$ is a positive constant. Then there exist a subsequence $\{f_{m_j}\}\subset\{f_j\}$ and a function $f\in L^1(I;\mathbb{R})$ such that $$ f_{m_j} \rightharpoonup f \quad\text{weakly in $L^1(I,\mathbb{R})$ as $j\to\infty$} $$ with $$ \int_I \Phi(f) dx \le \liminf_{j\to\infty} \int_I \Phi(f_{m_j}) dx \le C. $$ \end{lemma} Now we define the weak solution of problem \eqref{tv1}--\eqref{bd3}. \begin{definition} \label{def1} \rm Let $T$ be a fixed positive constant. A function $u: Q_T \to \mathbb{R}$ is called a weak solution of the problem \eqref{tv1}--\eqref{bd3}, if the following conditions are fulfilled: \begin{itemize} \item[(1)] $u\in C([0,T];L^2(I)) \cap L^{\infty}(0,T; W_0^{2,1}(I))$ and $\iint_{Q_T} \Phi\big(\frac{\partial^2 u}{\partial x^2}\big)\, dx\,dt < +\infty$. \item[(2)] For any $\varphi \in C_0^{\infty}(Q_T)$, $$ \iint_{Q_T} \big\{-u\frac{\partial \varphi}{\partial t} + \Phi'\big(\frac{\partial^2 u}{\partial x^2}\big)\frac{\partial^2 \varphi}{\partial x^2}\big\} dx\,dt = 0. $$ \item[(3)] $ u(x,0) = u_0(x)$ in $L^2(I)$. \end{itemize} \end{definition} We state our main result as follows. \begin{theorem} \label{theorem1} Let $u_0 \in L^2(I)$ with $\int_{I} \Phi(\frac{\partial^2 u_0}{\partial x^2})\, dx \le C$ and compatibility conditions on $\{0,1\}\times \{t=0\}$. Then problem \eqref{tv1}--\eqref{bd3} admits one and only one weak solution. \end{theorem} \section{Proof of the Main Theorem} We use the time discrete method to construct an approximate solution. Divide the interval $(0,T)$ into $N$ equal segments and denote $h=T/N$. Consider the problem: \begin{gather} \frac 1h\left( u_{k+1} - u_{k} \right) + \frac{d^2}{dx^2} \Phi'\big(\frac{d^2 u_{k+1}}{dx^2}\big) = 0, \label{lisan}\\ u_{k+1}(0) = u_{k+1}(1) = u'_{k+1}(0) = u'_{k+1}(1) = 0,\label{lbd1} \end{gather} where $k = 0,1,\dots, N-1$, and $u_0$ is the initial data. \begin{lemma}\label{lemlisan} For $u_k \in L^2(I)$, the problem \eqref{lisan}-\eqref{lbd1} admits one and only one weak solution $u_{k+1} \in W_0^{2,1}(I)$, such that for any $\phi(x)\in C_0^{\infty}(I)$, \begin{equation} \frac 1h\int_0^1\left(u_{k+1}-u_k\right)\phi dx + \int_0^1 \Phi'\big(\frac{d ^2 u_{k+1}}{d x^2}\big) \frac{d ^2 \phi}{d x^2} dx = 0, \label{lwas} \end{equation} and $$ \int_0^1 \Phi\big(\frac{d^2 u_{k+1}}{dx^2}\big) dx \le C, $$ where $C$ is a constant depended only on $\|u_k\|_{L^2(I)}$ and $h$. \end{lemma} \begin{proof} We investigate the functional defined on $W_0^{2,1}(I)$ by \begin{equation*} E (v) = \frac 1{2h}\int_0^1 (v - u_k)^2 dx + \int_0^1 \Phi\big(\frac {d^2v}{dx^2}\big) dx. \end{equation*} We choose $v = 0$, then $$ 0\le \inf_{v\in W_0^{2,1}(I)} E(v) \le E(0) = \frac 1{2h}\int_0^1 u_k ^2 dx. $$ By lemma \ref{con}, we can extract a minimizing sequence $\{v_n\}_{n=1}^\infty \subset W_0^{2,1}(I)$ such that $$ E(v_n) \to \inf_{v\in W_0^{2,1}(I)} E(v),\quad \text{as $n\to\infty$}, $$ and $$ \int_0^1 |v_n|^2 dx + \int_0^1 \Phi \big(\frac{d^2 v_n}{dx^2}\big) dx \le C. $$ By \eqref{cd1} and Lemma \ref{con}, we may find a subsequence $\{v_{n_j}\}_{j=1}^\infty\subset\{v_n\}_{n=1}^{\infty}$ and a function $u_{k+1}$, such that $v_{n_j} \rightharpoonup u_{k+1}$ weakly in $W_0^{2,1}(I)$ and $$ \int_0^1 \Phi\big(\frac{d^2u_{k+1}}{dx^2}\big) \le C. $$ Since $\Phi(s)$ is convex and by relaxation, we have that $u_{k+1}$ is a weak solution of the problem \eqref{lisan}--\eqref{lbd1}. Assume $u_{k+1}$ and $v_{k+1}$ are both solutions of the problem \eqref{lisan}--\eqref{lbd1}. Then for every $\phi(x)\in C_0^{\infty}(I)$, we have \begin{equation*} \frac 1h\int_0^1(u_{k+1}-v_{k+1})\phi dx + \int_0^1\Big( \Phi'\big(\frac{d ^2 u_{k+1}}{d x^2}\big) -\Phi'\big(\frac{d ^2 v_{k+1}}{d x^2}\big)\Big)\frac{d ^2 \phi}{d x^2} dx = 0. \end{equation*} By \eqref{ine2} and the approximation argument, we could take $\phi(x) = u_{k+1} - v_{k+1}$ as the test function. We get \begin{align*} &\frac 1h\int_0^1\left(u_{k+1}-v_{k+1}\right)^2 dx \\ &+ \int_0^1\Big( \Phi'\big(\frac{d ^2 u_{k+1}}{d x^2}\big) -\Phi'\big(\frac{d ^2 v_{k+1}}{d x^2}\big)\Big) \Big( \frac{d ^2 u_{k+1}}{d x^2}- \frac{d ^2 v_{k+1}}{d x^2}\Big)dx = 0. \end{align*} By \eqref{ine3}, the two terms on the left hand side are both nonnegative. We get $u_{k+1} = v_{k+1}$ a.e. in $I$. Then the proof is complete. \end{proof} Let $\chi^{h,j}(t)$ be the indicator function of $[h(j-1),hj)$. We construct an approximate function by \[ u^h(x,t) = \sum_{j=1}^{N}\chi^{h,j}(t)u_{j-1}(x) \quad \text{with}\quad u^h(x,0) = u_{0}(x). \] \begin{lemma}\label{lmgu1} For the weak solution $u_{k+1}$ of the problem \eqref{lisan}--\eqref{lbd1}, the following estimates hold \begin{gather}\label{es1} h\sum_{k=0}^{N-1} \int_0^1 \Phi'\left(\frac{d^2u_{k+1}}{dx^2}\right)\frac{d^2u_{k+1}}{dx^2} dx \le C, \\ \label{es2} \sup_{01$. Thus, we may extract a subsequence from $\xi_h$, denoted still by $\xi_h$, such that $$ \xi_h \rightharpoonup \xi \quad \text{weakly in $L^{p'}(\Omega)$}. $$ Since $\Psi(s)$ is a convex function, we obtain $$ \iint_{Q_T} \Psi(\xi) dx\,dt \le \liminf_{h\to 0} \iint_{Q_T}\Psi(\xi_h) dx\,dt \le C. $$ Using Young's inequality again, we have $$ \iint_{Q_T}\big|\xi\cdot \frac{\partial ^2 u}{\partial x^2}\big| dx\,dt \le \iint_{Q_T} \Psi(\xi) + \Phi\big(\frac{\partial ^2 u}{\partial x^2}\big)dx\,dt \le C. $$ By the discrete equation \eqref{liss}, we see that $$ \frac 1h \Delta^h u^h \quad\text{is bounded in $L^\infty(0,T;W^{-2,1}(I))$} $$ and $$ \frac 1h \Delta^h u^h \rightharpoonup \frac{\partial u}{\partial t} \quad\text{weakly * in $L^\infty(0,T;W^{-2,1}(I))$}. $$ Letting $h\to 0$ in \eqref{liss}, we have in the sense of distributions \begin{equation}\label{dis} \frac{\partial u}{\partial t} + \frac{\partial^2 \xi}{\partial x^2} = 0. \end{equation} Now we will prove $\xi = \Phi'\big(\frac{\partial^2 u}{\partial x^2}\big)$. Denote $$ f_h(t) = \frac{t-kh}{2h}\Big(\int_0^1 |u_{k+1}|^2dx - \int_0^1 |u_k|^2dx\Big) + \frac 12 \int_0^1 u_k^2dx, $$ where $kh < t \le (k+1)h$, $k=0,1,2,\dots, N-1$. By \eqref{lem5}, we have \begin{gather*} \frac 12 \int_0^1 |u_k|^2 dx - Ch \le f_h(t) \le \frac 12 \int_0^1 |u_k|^2dx,\\ -C \le f'_h(t) \le 0. \end{gather*} According to the Ascoli-Arzela theorem, there exists a function $f(t) \in C([0,T])$, such that $$ \lim_{h\to 0}f_h(t) = \frac 12\lim_{h\to 0}\int_0^1 |u^h|^2 dx = f(t) \quad\text{uniformly for $t\in[0,T]$}. $$ It follows from \eqref{tmp1} that $$ \frac 12\int_0^1 |u^h|^2 dx + \iint_{Q_T} \Phi'\big(\frac{\partial ^2 u^h}{\partial x^2}\big)\frac{\partial ^2 u^h}{\partial x^2} dx\,dt \le \frac 12 \int_0^1 |u_0|^2 dx. $$ Letting $h\to 0 $ in the above inequality we have \begin{align*} &\liminf_{h\to 0} \iint_{Q_T} \Phi'\left(\frac{\partial ^2 u^h}{\partial x^2}\right) \frac{\partial ^2 u^h}{\partial x^2} dx\,dt \\ &\le f(0)- f(T) \\ &= \lim_{\varepsilon\to 0^+} \frac 1\varepsilon \int_0^{T-\varepsilon} (f(t)-f(t+\varepsilon)) dt \\ &= \lim_{\varepsilon\to 0^+} \lim_{h \to 0} \frac 1{2\varepsilon} \int_0^{T-\varepsilon} \int_0^1 (|u^h(x,t)|^2 - |u^h(x,t+\varepsilon)|^2) dx\,dt \\ &\le \lim_{\varepsilon\to 0^+} \frac 1\varepsilon \int_0^{T-\varepsilon}\int_0^1 (u(x,t)-u(x,t+\varepsilon))\cdot u\,dx\,dt\\ &\le -\int_0^T \langle \frac{\partial u}{\partial t},u \rangle dt, \end{align*} where $\langle \cdot\rangle $ denotes the dual product of the function in $W^{-2,1}(I)$ and $W_0^{2,1}(I)$. So we have \begin{equation}\label{lsc} \liminf_{h\to 0} \iint_{Q_T} \Phi'\big(\frac{\partial ^2 u^h}{\partial x^2}\big)\frac{\partial ^2 u^h}{\partial x^2} dx\,dt \le \iint_{Q_T} \xi \frac{\partial^2 u}{\partial x^2} dx\,dt. \end{equation} Define $F[u] = \int_0^1 \Phi\big(\frac{\partial^2 u}{\partial x^2}\big) dx$ and choose a function $g\in L^\infty(0,T; W_0^{2,1}(I))$ with $\iint_{Q_T} \Phi\big(\frac{\partial^2 g}{\partial x^2}\big)\, dx\,dt < +\infty$. Because $\Phi(s)$ is convex, we have $$ \iint_{Q_T} \Phi\big(\frac{\partial^2 g}{\partial x^2}\big) dx\,dt - \iint_{Q_T} \Phi\big(\frac{\partial^2 u^h}{\partial x^2}\big)dx\,dt \ge \iint_{Q_T}\Phi'\big(\frac{\partial^2 u^h}{\partial x^2}\big) \frac{\partial^2 (g -u^h)}{\partial x^2} dx\,dt. $$ Letting $h\to 0$ and by \eqref{lsc}, we get $$ \iint_{Q_T} \Phi\big(\frac{\partial^2 g}{\partial x^2}\big) dx\,dt - \iint_{Q_T} \Phi\big(\frac{\partial^2 u}{\partial x^2}\big) dx\,dt \ge \iint_{Q_T} \xi\cdot\frac{\partial^2 (g -u)}{\partial x^2} dx\,dt. $$ Replacing $g$ by $\varepsilon g + u$, we see that $$ \frac 1\varepsilon(F[u+\varepsilon g]-F[u]) \ge \iint_{Q_T} \xi\cdot \frac{\partial^2 g}{\partial x^2}dx\,dt $$ and $$ \iint_{Q_T} \frac{\delta F[u]}{\delta u} g dx\,dt = \iint_{Q_T} \Phi'\big(\frac{\partial^2 u}{\partial x^2}\big) \frac{\partial ^2 g}{\partial x^2} dx\,dt \ge \iint_{Q_T} \xi \cdot \frac{\partial ^2 g}{\partial x^2} dx\,dt. $$ Due to the arbitrariness of $g$, we get that $\xi = \Phi'\big(\frac{\partial^2 u}{\partial x^2}\big)$. By \eqref{dis}, $u$ is the weak solution of the problem \eqref{tv1}--\eqref{bd3}. Next, we prove the uniqueness of the weak solution of the problem \eqref{tv1}--\eqref{bd3}. Suppose there exist two weak solutions $u$ and $v$. Using an approximation technique (see \cite{wzyl,XZ}), for any test function $\varphi(x,t)\in C^{\infty}(\bar Q_T)$, we have $$ \iint_{Q_T} -(u-v)\frac{\partial \varphi}{\partial t} dx\,dt + \iint_{Q_T} \Big(\Phi'\big(\frac{\partial^2 u}{\partial x^2}\big) - \Phi'\big(\frac{\partial^2 v}{\partial x^2}\big)\Big)\frac{\partial^2 \varphi}{\partial x^2} dx\,dt = 0. $$ Furthermore, we may take $u-v$ as a test function and then get $$ \frac 12\int_0^1 |u-v|^2(t) dx\,dt + \iint_{Q_t} \Big(\Phi'\big(\frac{\partial^2 u}{\partial x^2}\big) - \Phi'\big(\frac{\partial^2 v}{\partial x^2}\big)\Big)\Big(\frac{\partial^2 u}{\partial x^2} - \frac{\partial^2 v}{\partial x^2} \Big) dx\,dt = 0, $$ where $Q_t = (0,t)\times I$. Since the two terms on the left hand side are nonnegative by inequality \eqref{ine4}, we have $u=v$ a.e. in $Q_T$. Thus the proof is complete. \end{proof} \section{Numerical experiments} After the theoretical analysis, we shall do some numerical tests of higher order filters in practice to compare our model with the other well-known models of \cite{pm,you}. In our model, we take $ \Phi(s) = |s|\ln(1+|s|)$. For convenience, we are in favor of implementation of an explicit Euler method, i.e. $$ \frac{u^{k+1} - u^k}{\Delta t} + \frac{\partial ^2}{\partial x^2} \Phi'\big(\frac{\partial ^2 u^k}{\partial x^2}\big)= 0, \quad(x,t) \in Q_T. $$ For each figure, we use $1$ for space steps, $0.2$ for time steps of figure (c) and $0.001$ for time steps of figures (d) and (e). Steady state was achieved for figure (d) and figure (e) in less than $20000$ iterations. In figure (c), we fixed the number of iterations to $1500$. Fig.1 (a) shows the initial signal and (b) the noisy signal. By the figures from (c) to (e), we could conclude that the second order filtering yields enhancement of edges and staircase-like structures, the fourth order filtering results tend to be piecewise linear with enhanced curvature. At the same time we could also see that the fourth order filtering is further affirmed by the almost piecewise constant derivative which is also shown in Figure 1. \begin{figure}[ht] \begin{center} \includegraphics[width=0.45\textwidth]{fig1a} \includegraphics[width=0.45\textwidth]{fig1b} \\ \includegraphics[width=0.45\textwidth]{fig1c} \includegraphics[width=0.45\textwidth]{fig1d} \\ \includegraphics[width=0.45\textwidth]{fig1e} \end{center} \caption{One-dimensional signal evaluation: original signal, noisy signal, and restored by second order Perona-Malik model, fourth-order Perona-Malik model and our model.} \end{figure} \subsection*{Acknowledgements} The authors would like to express their sincerely thanks to Prof. J.X. 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